$$
\begin{array}{l} \mathcal {L} (w _ {t + 1}) \leq \mathcal {L} (w _ {t}) - \alpha \left\langle \operatorname {g r a d} \mathcal {L} (w _ {t}), \mathcal {T} (\gamma) _ {w _ {t} ^ {a d v}} ^ {w _ {t}} \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t} ^ {a d v}) \right\rangle_ {w _ {t}} + \frac {\alpha^ {2} L _ {S}}{2} \left\| \mathcal {T} (\gamma) _ {w _ {t} ^ {a d v}} ^ {w _ {t}} \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t} ^ {a d v}) \right\| _ {w _ {t}} ^ {2} \\ = \mathcal {L} (w _ {t}) - \frac {\alpha^ {2} L _ {S}}{2} \| \operatorname {g r a d} \mathcal {L} (w _ {t}) \| _ {w _ {t}} ^ {2} + \frac {\alpha^ {2} L _ {S}}{2} \left\| \mathcal {T} (\gamma) _ {w _ {t} ^ {a d v}} ^ {w _ {t}} \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t} ^ {a d v}) - \operatorname {g r a d} \mathcal {L} (w _ {t}) \right\| _ {w _ {t}} ^ {2} \\ - \alpha (1 - \alpha L _ {S}) \Big \langle \mathcal {T} (\gamma) _ {w _ {t} ^ {a d v}} ^ {w _ {t}} \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t} ^ {a d v}), \operatorname {g r a d} \mathcal {L} (w _ {t}) \Big \rangle_ {w _ {t}} \\ \leq \mathcal {L} (w _ {t}) - \frac {\alpha^ {2} L _ {S}}{2} \| \operatorname {g r a d} \mathcal {L} (w _ {t}) \| _ {w _ {t}} ^ {2} + \alpha^ {2} L _ {S} \left\| \mathcal {T} (\gamma) _ {w _ {t} ^ {a d v}} ^ {w _ {t}} \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t} ^ {a d v}) - \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t}) \right\| _ {w _ {t}} ^ {2} \\ + \alpha^ {2} L _ {S} \| \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t}) - \operatorname {g r a d} \mathcal {L} (w _ {t}) \| _ {w _ {t}} ^ {2} - \alpha (1 - \alpha L _ {S}) \Big \langle \mathcal {T} (\gamma) _ {w _ {t} ^ {a d v}} ^ {w _ {t}} \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t} ^ {a d v}), \operatorname {g r a d} \mathcal {L} (w _ {t}) \Big \rangle_ {w _ {t}} \\ \leq \mathcal {L} (w _ {t}) - \frac {\alpha^ {2} L _ {S}}{2} \| \operatorname {g r a d} \mathcal {L} (w _ {t}) \| _ {w _ {t}} ^ {2} + \alpha^ {2} L _ {S} ^ {3} \rho^ {2} \| \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t}) \| _ {w _ {t}} ^ {2} + \alpha^ {2} L _ {S} \| \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t}) - \operatorname {g r a d} \mathcal {L} (w _ {t}) \| _ {w _ {t}} ^ {2} \\ - \alpha (1 - \alpha L _ {S}) \Big \langle \mathcal {T} (\gamma) _ {w _ {t} ^ {a d v}} ^ {w _ {t}} \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t} ^ {a d v}), \operatorname {g r a d} \mathcal {L} (w _ {t}) \Big \rangle_ {w _ {t}} \\ \leq \mathcal {L} (w _ {t}) - \frac {\alpha^ {2} L _ {S}}{2} \| \operatorname {g r a d} \mathcal {L} (w _ {t}) \| _ {w _ {t}} ^ {2} + 2 \alpha^ {2} L _ {S} ^ {3} \rho^ {2} \| \operatorname {g r a d} \mathcal {L} (w _ {t}) \| _ {w _ {t}} ^ {2} + 2 \alpha^ {2} L _ {S} ^ {3} \rho^ {2} \| \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t}) - \operatorname {g r a d} \mathcal {L} (w _ {t}) \| _ {w _ {t}} ^ {2} \\ + \alpha^ {2} L _ {S} \| \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t}) - \operatorname {g r a d} \mathcal {L} (w _ {t}) \| _ {w _ {t}} ^ {2} - \alpha (1 - \alpha L _ {S}) \Big \langle \mathcal {T} (\gamma) _ {w _ {t} ^ {a d v}} ^ {w _ {t}} \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t} ^ {a d v}), \operatorname {g r a d} \mathcal {L} (w _ {t}) \Big \rangle_ {w _ {t}} \\ = \mathcal {L} \left(w _ {t}\right) - \frac {\alpha^ {2} L _ {S} \left(1 - 4 L _ {S} ^ {2} \rho^ {2}\right)}{2} \| \operatorname {g r a d} \mathcal {L} \left(w _ {t}\right) \| _ {w _ {t}} ^ {2} + \alpha^ {2} L _ {S} \left(1 + 2 L _ {S} ^ {2} \rho^ {2}\right) \| \operatorname {g r a d} \mathcal {L} _ {t} \left(w _ {t}\right) - \operatorname {g r a d} \mathcal {L} \left(w _ {t}\right) \| _ {w _ {t}} ^ {2} \\ - \alpha (1 - \alpha L _ {S}) \Big \langle \mathcal {T} (\gamma) _ {w _ {t} ^ {a d v}} ^ {w _ {t}} \operatorname {g r a d} \mathcal {L} _ {t} (w _ {t} ^ {a d v}), \operatorname {g r a d} \mathcal {L} (w _ {t}) \Big \rangle_ {w _ {t}} \\ \end{array}
$$